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Properties

Label 1-311-311.270-r0-0-0
Degree $1$
Conductor $311$
Sign $0.945 + 0.326i$
Analytic cond. $1.44427$
Root an. cond. $1.44427$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.758 + 0.651i)2-s + (−0.758 − 0.651i)3-s + (0.151 − 0.988i)4-s + (0.820 − 0.571i)5-s + 6-s + (−0.612 − 0.790i)7-s + (0.528 + 0.848i)8-s + (0.151 + 0.988i)9-s + (−0.250 + 0.968i)10-s + (0.347 + 0.937i)11-s + (−0.758 + 0.651i)12-s + (0.151 + 0.988i)13-s + (0.979 + 0.201i)14-s + (−0.994 − 0.101i)15-s + (−0.954 − 0.299i)16-s + (−0.250 + 0.968i)17-s + ⋯
L(s)  = 1  + (−0.758 + 0.651i)2-s + (−0.758 − 0.651i)3-s + (0.151 − 0.988i)4-s + (0.820 − 0.571i)5-s + 6-s + (−0.612 − 0.790i)7-s + (0.528 + 0.848i)8-s + (0.151 + 0.988i)9-s + (−0.250 + 0.968i)10-s + (0.347 + 0.937i)11-s + (−0.758 + 0.651i)12-s + (0.151 + 0.988i)13-s + (0.979 + 0.201i)14-s + (−0.994 − 0.101i)15-s + (−0.954 − 0.299i)16-s + (−0.250 + 0.968i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.945 + 0.326i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (270, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.945 + 0.326i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7199443969 + 0.1206862181i\)
\(L(\frac12)\) \(\approx\) \(0.7199443969 + 0.1206862181i\)
\(L(1)\) \(\approx\) \(0.6795932439 + 0.04298389292i\)
\(L(1)\) \(\approx\) \(0.6795932439 + 0.04298389292i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.758 + 0.651i)T \)
3 \( 1 + (-0.758 - 0.651i)T \)
5 \( 1 + (0.820 - 0.571i)T \)
7 \( 1 + (-0.612 - 0.790i)T \)
11 \( 1 + (0.347 + 0.937i)T \)
13 \( 1 + (0.151 + 0.988i)T \)
17 \( 1 + (-0.250 + 0.968i)T \)
19 \( 1 + (0.820 + 0.571i)T \)
23 \( 1 + (0.528 + 0.848i)T \)
29 \( 1 + (0.979 - 0.201i)T \)
31 \( 1 + (-0.250 - 0.968i)T \)
37 \( 1 + (-0.612 + 0.790i)T \)
41 \( 1 + (0.688 + 0.724i)T \)
43 \( 1 + (-0.612 - 0.790i)T \)
47 \( 1 + (0.979 - 0.201i)T \)
53 \( 1 + (-0.612 - 0.790i)T \)
59 \( 1 + (-0.612 - 0.790i)T \)
61 \( 1 + (0.820 + 0.571i)T \)
67 \( 1 + (0.688 - 0.724i)T \)
71 \( 1 + (0.918 + 0.394i)T \)
73 \( 1 + (-0.440 + 0.897i)T \)
79 \( 1 + (0.688 + 0.724i)T \)
83 \( 1 + (-0.994 + 0.101i)T \)
89 \( 1 + (-0.612 + 0.790i)T \)
97 \( 1 + (0.918 + 0.394i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.28877334995633722393076325621, −24.70723292823970392892456778377, −22.84088019427707717710906024026, −22.28010671592782030637199516733, −21.66757111447926549573648334487, −20.883674341591573237283516430325, −19.792213697689376065684596888740, −18.616258417671982342289669279097, −18.04275731411322064580611005497, −17.28008234408864620024833705548, −16.16843624199507107985377806193, −15.63539503478265690732371714017, −14.137079671661268707213940702200, −12.935073727338244071119170080589, −12.00738823599710440828188315704, −11.00831629112162753221015562442, −10.38859153973540976392787839853, −9.36609931161660927008317763565, −8.811856065185448658377548894845, −7.04226816700059168964316917158, −6.12343117814222620510673715056, −5.07992247770597016061041955807, −3.31778365581601187421884203016, −2.731528014513371600314691661814, −0.83554400744856216639503666027, 1.13803836503766234763089671661, 1.93475801536947595425068757393, 4.36237686460336751124648719954, 5.48462678322724061965963129322, 6.473774966902552855277511668183, 7.04183298565100412390228043446, 8.21144112990133996632951505165, 9.53587496025611309852333740033, 10.07562625740374356474362077762, 11.24710697636693315269204200624, 12.42979660938507759895400429067, 13.46862779225039513793225118895, 14.15275985527683281457870647687, 15.63017033924322816368505165830, 16.673849625448332594691627418222, 17.0970158432783799584212435605, 17.77395486875386614855991561088, 18.813847406733282227912134082052, 19.65619679434170147087176629170, 20.55352811885161942386029210771, 21.921400156544840213009094223504, 22.96802633849536753651712372033, 23.673392627176458854900313386229, 24.42964975836798073706106070295, 25.353249712140922355497925410175

Graph of the $Z$-function along the critical line